Math-in-CTE Lesson Plan Lesson Title: Maximizing Profits Lesson # 7 Author(s): Phone Number(s): E-mail Address(es): Skip Walker 720-423-6100 James_walker@dpsk12.org Dana Starbuck Occupational Area: Business CTE Concept(s): Break Even Analysis Math Concepts: Solving Algebraic Equations & Linear Programming Lesson Objective: After completion of this lesson, the student should be able to compute low level breakeven points and complex breakeven points using linear programming Supplies Needed: Linear Programming Paper Linear Programming Worksheet Worksheet Answer Key Graphing Review Sheet (if needed) THE "7 ELEMENTS" TEACHER NOTES (and answer key) 1. Introduce the CTE lesson. Discuss the definition of: Many manufacturers produce several Maximize: To increase or make as products, they need to make enough of each great as possible or to find the largest product to meet their customers need and value of a function best utilize their use of materials, labor, and Linear Programming: A mathematical machinery. technique used in economics; finds the As a manufacturer you need to decide which maximum or minimum of linear functions products to produce that will maximize your in many variables subject to constraints profits. This is done through a process Break-even Point: The point, especially called Linear Programming the level of sales of a good or service, at Before we can discuss maximum profits, we which the return on investment is exactly must first find out when the manufacturer will equal to the amount invested. begin making a profit. This is called the *Break-even point has previously been Break-even Point discussed* 2. Assess students’ math awareness as it Break-even Point: Income=Expenses relates to the CTE lesson. Equation: 3x + 18,000 = 15x To assess student’s math background, give a -3x -3x short warm-up quiz with questions relating to 18,000 = 12x Break-even. 12 12 What is the formula for Break-even Point? X = 1500 Try this problem: We are making widgets: That is, 1500 Widgets need to be sold to variable costs $3 per widget; fixed costs total make a profit. $18,000; and we are selling them for $15 each. What would the breakeven point be? 3. Work through the math example embedded Linear Programming Paper has been in the CTE lesson. included; it is nice to be able to do the Otto Toyom builds toy cars and toy trucks. problem at the top of the page, while Each car needs 4 wheels, 2 seats, and 1 gas having the graph paper included. tank. Each truck needs 6 wheels, 1 seat, and Constraint: The state of being restricted 3 gas tanks. His storeroom has 36 wheels, or confined within prescribed bounds. 14 seats, and 15 gas tanks. He makes $1.00 on each car and $1.00 on each truck he KEY: sells. 2. Wheels [4x + 6y ≤ 36] What combination of cars and trucks will In English, this equation says that each maximize Otto’s profit? car needs 4 wheels, each truck needs 6 Steps we must take to solve this problem: wheels and together they can not add to be more than 36 (which is the number of 1. Pick variables. wheels Otto can hold in his storeroom). Let x = # cars and y = # trucks. Seats [2x + y ≤ 14] 2. Write inequalities that model each constraint. Gas tanks [x + 3y ≤ 15] 3. Graph each constraint. 3. Graphing inequalities will need to be reviewed with students. (See attached 4. Calculate intersections of each inequality. sheet for further review). (Using Systems of Equations) Students will also need to understand 5. Write a Profit Equation. why we are looking at graphs only in the 6. Substitute each ordered pair from Step #4 first quadrant (where x and y are both into the profit equation from step #5. positive). Explain that there is no way for Determine which ordered pair has the Otto to make a negative number of either highest profit. This is the Linear Combination cars or trucks, thus x and y must both be that maximizes the profit! positive. 4. Students must know that ANY maximum or minimum will occur at one of the vertices. (These are the intersections of the inequalities). Solving Systems of Equations will need to be reviewed with students. (See attached for further review). Intersections: (6, 2), (3, 4), (5.4, 3.2) use (5, 3) since you can’t make a fraction of a car or truck. 5. Profit = 1x + 1y 6. Max Profit: (6,2) Profit = $8 (3, 4) Profit = $7 (5, 3) Profit = $8 So Max occurs either with 6 cars and 2 trucks or 5 cars and 3 trucks. 4. Work through related, contextual math-in- KEY: CTE examples. Profit = 1x + 2y Truck drivers have just become popular because of a new TV series called “Big Red (6,2) Profit = $10 Ed.” Toy trucks are a hot item. Otto can now (3, 4) Profit = $11 make $2.00 per truck though he stills gets (5, 3) Profit = $11 $1.00 per car. He has hired you as a consultant to advise him, and your salary is a So now, the Max occurs either with 3 cars percentage of the total profits. What is his and 4 trucks or 5 cars and 3 trucks. best choice for the number of cars and the number of trucks to make now? How can you be sure? Explain. 5. Work through traditional math examples. USE LINEAR PROGRAMMING PAPER The Smiths have a small farm where they FOR WORK. grow corn and tomatoes for sale. They have 1. Let x = the number of acres of corn a total of 21 acres available for planting. planted and y = the number of acres of Because they cannot afford to pay a lot for tomatoes planted. help, they have many restrictions based on 2. Write inequalities which represent the their labor. They have a total of 150 hours restrictions on the land available. (x + y ≤ available for planting and 130 hours available 21) for harvesting. Each acre of corn takes 6 The time available for planting: hours to plant and 4 hours to harvest. Each (6x + 10y ≤ 150) acre of tomatoes takes 10 hours to plant and The time available for harvesting: 10 hours to harvest. A local grocery chain (4x + 10y ≤ 130) has agreed to purchase 3 acres worth of The arrangement made with the grocery tomatoes. chain: (y ≥ 3) If each acre of corn can be sold for $600 and 3-4.Graph the system you found in part each acre of tomatoes can be sold for $800, (a) and find the vertices: (Vertices (x , how many acres of each type should the y): (0, 3); (18, 3): (15, 6); (10, 9); (0, 13)) Smiths plant? 5. Profit = 600x + 800y 6. Substitute each vertex into the Profit equation: (0,3) Profit = $2,400 (18,3) Profit = $13,200 **MAX**(15,6) Profit = $13,800 (10,9) Profit = $13,200 (0, 13) Profit = $10,400 6. Students demonstrate their understanding. See Answer Sheets for completed See Attached Linear Programming Graphs. Worksheet Use with Linear Programming ANSWERS: Paper (This assignment may take several OIL: nights as the problems can be challenging) a. Profit=30T + 15C b. T + C ≤ 40000; T + C ≥ 18000; 6C + 2T ≤ 120000 c. See graph d. (Texas, Calif.) = (40,000, 0) Profit = $1200000 PARK: a. x=old member; y=new member Profit =10x + 8 y b. x ≥ 0, y ≥ 0; x ≤ 9, y ≤ 8; 6 ≤ x + y ≤ 1 15, y ≥ 3; y ≥ x, y ≤ 3x 2 c. See graph d. No, you must have old members because the graph isn’t shaded at x = 0. e. (old, new) = (9,6) Profit = $138 f. (old, new) = (1.5, 4.5) but since we can’t have 0.5 of a person, use (2, 4) Profit = $52 g. Profit=10x + 12y (old, new) = (7, 8) Profit= $166 AIRCRAFT: a. x = Camel, y = Hippo Profit = 300x + 200y b. x + y ≤ 12; x ≤ 11, y ≤ 7; y ≤ 2x; 100x + 200y ≥ 1000 c. See graph (Camel, Hippo) = (11, 1) Profit = $3500 7. Formal assessment. ANSWER: You are working at a small art store and your X = Pastel Y= Watercolor boss has decided to allow you sell some of Profit = 40x + 100y your art. Your boss has determined that you Constraints: 5x + 15y ≤ 180; x + y ≤ 16; can choose what combination of art you x ≥ 3, y ≥ 2 would like to sell, so that you can make the most profit possible. Possible Maximum’s: (3, 11); (3, 2); (6, 10); (14, 2) 1. Each pastel requires $5 in materials and earns a profit of $40. 2. Each watercolor requires $15 in materials MAX: (6, 10) Profit =$1240 and earns a profit of $100. 3. You have $180 to spend on materials. 4. You plan to make at least 3 pastels and at least 2 watercolors. 5. You can make at most 16 pictures. What is the optimum number of pastels and watercolors that produces the maximum profit?